Matrices; 11.2

​We will be entering The Matrix

Matrices; 11.2

A matrix is a bracket of elements also called cells. The cells are listed in order of rows by columns. 2X3 would be a matrix with 2 rows and 3 columns

​Matrices are used to solve systems of equations.
Lets first learn how we write them as there is an organization to them. Nothing is random.

Multiple Choice

$$a_{33}$$  This notation is used for cell location where r=row and c=column.  What value is in the following location?

$$a_{rc}$$  

• Yes: -12

• not 5

• not -3

• not zero

Multiple Choice

$$a_{21}$$  This notation is used for cell location where r=row and c=column.  What value is in the following location?

Reminder : $$a_{rc}$$  

• -12

• 9

• 6

• Zer0

Multiple Choice

$$a_{13}$$  This notation is used for cell location where r=row and c=column.  What value is in the following location?

Reminder : $$a_{rc}$$  

• 13

• -2

• 6

• Zer0

Multiple Choice

What are the dimensions of this matrix?

• 2x3

• 2x2

• 3x3

• 3x2

Multiple Choice

Determine the dimensions of Matrix C

• 1x1

• 1x4

• 4x1

• 4x4

Multiple Choice

What is in $$a_{21}$$

• p

• q

• r

• s

We use matrices to help in solving systems of equations

Here is an example of a coefficient matrix. You will have an x column and a y column and their sign must come with.

Multiple Choice

Which of these is the correct coefficient matrix?

Augmented Matrix

An augmented matrix will have the coefficients as well as the constants. As you can see the constants are separated by a line.

​*Please notice that a matrix has [ <--- brackets, this is important.
Sometimes the line in the middle of an augmented matrix is left out.

Multiple Choice

Which system below represents this augmented matrix?

Multiple Select

A coefficient matrix includes the constants?

• True

• False

Multiple Select

The constants of the systems of equations are included with an augmented matrix?

• true

• false

Multiple Choice

The second row of this augmented matrix could have one of the following equations.

• $$2x+3y-z=-2$$  

• $$-x+3y-2z=8$$  

• $$x-y+1=8$$  

• $$3x-2y-9=9$$  

Definition of Row Echelon Form (REF)
>which is a method of solving a system using matrices

the matrix would look like this.....

Ones on the diagonal; top left to bottom right of portion of values that are the coefficients.

• a, b, c, d, e, and f are real numbers

• the last row would tell us that the value for z is equal to the value of f. Or simply 1z=f

Multiple Select

What would we get for y with this information?

$$y+5z=11\ and\ z=2$$      

• 1

• 10

• 21

• -1

Now that you have z=2 and y=1; use these outcomes to sub into the first row and solve for x. Do this before going to the next few slides.

Multiple Choice

Determine the equation using the first row.

• $$x+3y-2z=-3$$

• $$x+5y=11$$

• $$x+3y+2z=-3$$

Multiple Choice

Using this information what is the value for x?

• x=-2

• x=2

• x=-8

• x=1

Multiple Choice

Use this matrix to determine the values of x, y & z

• -9, 1/2, -1

• -9, 5/2, -1

• 9, 1/2, -1

• 9, 5/2, -1

Multiple Choice

Use this to determine the values of x,y & z (with the understanding that the columns are in the order of x, y, & z)

• 3, -2, 7

• 7, -2, 3

• -2, 3, 7

• 1,1,1

That last matrix of a system of equations was amazingly easy to obtain the unknown values for the variables. Why?

This structure is called Reduced Row Echelon Form. Also known as; RREF

Time For......

• A matrix is a rectangular list of data.

• The dimensions are row X column

• REF; Row Echelon Form. 1's on the diagonals, zeros below those 1's, any numbers elsewhere.

• RREF: Reduced Row Echelon Form. 1's on the diagonals zeros above and below those 1's and a column of values on the far right

How do we get a matrix into REF --->
or RREF?

For REF; Goal is to get 1's on the diagonal and zeros below each 1. This is important!

• Move any row to another row

• Add a row to another row and replace a row

• Take a multiple of a row and add it to another row and replace a row

• Divide an entire row by the same number

• We will start with row 1 and work through the columns from left to right...

System to Augmented Matrix

We should now have zeros in under the first 1.

• We will now move to column 2 and figure out how to get a 1 in the center spot.

• What do you think is the easiest way to get 1 in the center?

Multiple Choice

I say do this.....

• divide all of row 2 by 2

• no other choice would be better! : )

Notation for that move is...

Here is the next matrix

Please notice we rewrite the entire other elements in the entire matrix.

This next step you must use row 2 (not row 1) to get a zero to replace the 5. If you use row 1 now you will end up with values back into the first column & we do not want that!

Multiple Choice

$$5r_2-r_3\rightarrow R_3$$  puts these values into row 3 respectively. 

• 0  0  -2  -6

• 0  0  2  -6

• 0  0  -2  6

• 0  10  -2  -4

So we rewrite the entire matrix and what is our last step?

​

Get's this final outcome

REF; Row Echelon Form!!!

We can clearly see 1z=3, or just z=3

Take this value right now and find the values for x and y to answer the next slide.

Multiple Choice

The values for x and y are....

• x=1 & y=2

• x=2 & y=1

• x=1 & y=3

• x=3 & y=2

Needed Definitions

Consistent: there is a solution

Inconsistent: there is no solution

Independent: one unique solution

Dependent: Infinitely many solutions: when you allow one variable to be "any value" and the other variables are in terms of that variable.

Summary

• A matrix is another way to write out a system of equations without the variables

• Using the matrix you perform row operations to get 1's on the diagonal and zeros below them

• we must rewrite each matrix as there is tons of room for errors

• On another slide there will be a video for RREF.

Multiple Choice

The ordered quad would be, (w,x,y,z), with w in the 1st column and z in the 4th column? You could even have $$\left(a_1,\ a_2,\ a_3,\ a_4\right)$$ But the outcome would be not an ordered pair or triple but an ordered quad.

• (2,-3,5,7)

• (1,1,1,1)

• (1, -3, 1, 5)

• (0,0,0,7)

Write this system in augmented form and then perform row operations to get in REF form...

solve & back sub to get all solutions.

THEN work on putting the REF in RREF form (since you back-subed you know the solutions but need to practice RREF form)

Good Job!